Normal modal logic
In logic, a normal modal logic is a set L of modal formulas such that L contains:
- All propositional tautologies;
- All instances of the Kripke schema:
- Detachment rule : ;
- Necessitation rule: implies.
Every normal modal logic is regular and hence classical.
Common normal modal logics
The following table lists several common normal modal systems.
Name | Axioms | Frame condition |
K | — | all frames |
T | T | reflexive |
K4 | 4 | transitive |
S4 | T, 4 | preorder |
S5 | T, 5 or D, B, 4 | equivalence relation |
S4.3 | T, 4, H | total preorder |
S4.1 | T, 4, M | preorder, |
S4.2 | T, 4, G | directed preorder |
GL, K4W | GL or 4, GL | finite strict partial order |
Grz, S4Grz | Grz or T, 4, Grz | finite partial order |
D | D | serial |
D45 | D, 4, 5 | transitive, serial, and Euclidean |